Question: Simplify and expand the following expression: $ \dfrac{p - 5}{5p + 8}-\dfrac{p - 1}{5p - 4} $
Answer: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5p + 8)(5p - 4)$ Multiply the first term by $\dfrac{5p - 4}{5p - 4}$ $ \begin{align*} \dfrac{p - 5}{5p + 8} \times \dfrac{5p - 4}{5p - 4} & = \dfrac{(p - 5)(5p - 4)}{(5p + 8)(5p - 4)} \\ & = \dfrac{5p^2 - 29p + 20}{(5p + 8)(5p - 4)}\end{align*} $ Multiply the second term by $\dfrac{5p + 8}{5p + 8}$ $ \begin{align*} \dfrac{p - 1}{5p - 4} \times \dfrac{5p + 8}{5p + 8} & = \dfrac{(p - 1)(5p + 8)}{(5p - 4)(5p + 8)} \\ & = \dfrac{5p^2 + 3p - 8}{(5p - 4)(5p + 8)}\end{align*} $ Now we have: $ = \dfrac{5p^2 - 29p + 20}{(5p + 8)(5p - 4)} - \dfrac{5p^2 + 3p - 8}{(5p - 4)(5p + 8)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{5p^2 - 29p + 20 - (5p^2 + 3p - 8)}{(5p + 8)(5p - 4)} $ $ = \dfrac{5p^2 - 29p + 20 - 5p^2 - 3p + 8}{(5p + 8)(5p - 4)} $ $ = \dfrac{-32p + 28}{(5p + 8)(5p - 4)}$ Expand the denominator: $ = \dfrac{-32p + 28}{25p^2 + 20p - 32}$